### 5.2 Energy principles

Energy principles leading to various magnetic fields (potential fields, linear force-free fields, and nonlinear force-free fields) were summarized in Sakurai (1989). For a given distribution of magnetic flux () on the boundary,
• a potential field is the state of minimum energy.
• If the magnetic energy is minimized with an additional condition of a fixed value of , one obtains a linear force-free field. The value of constant should be an implicit function of . The obtained solution may or may not be a minimum of energy; in the latter case the solution is dynamically unstable.
• If the magnetic energy is minimized by specifying the connectivity of all the field lines, one obtains a nonlinear force-free field. The solution may or may not be dynamically stable.

Item (c) is more explicitly shown by introducing the so-called Euler potentials for the magnetic field (Stern, 1970),

This representation satisfies . Since , and are constant along the field line. The values of and on the boundary can be set so that matches the given boundary condition. If the magnetic energy is minimized with the values of and specified on the boundary, one obtains Equation (5) for a general (nonlinear) force-free field.

By the construction of the energy principles, the energy of (b) or (c) is always larger than that of the potential field (a). If the values of and are so chosen (there is enough freedom) that the value of is the same in cases (b) and (c), then the energy of nonlinear force-free fields (c) is larger than that of the linear force-free field (b). Therefore, we have seen that magnetic energy increases as one goes from a potential field to a linear force-free field, and further to a nonlinear force-free field. Suppose there are field lines with enhanced values of (carrying electric currents stronger than the surroundings). By some instability (or magnetic reconnection), the excess energy may be released and the twist in this part of the volume may diminish. However, in such rapid energy release processes, the magnetic helicity over the whole volume tends to be conserved (Berger, 1984). Namely local twists represented by spatially-varying only propagate out from the region and are homogenized, but do not disappear. Because of energy principle (b), the end state of such relaxation will be a linear force-free field. This theory (Taylor relaxation; Taylor, 1974, 1986) explains the commonly-observed behavior of laboratory plasmas to relax toward linear force-free fields. On the Sun this behaviour is not observed, however. A possible explanation could be that since we observe spatially-varying on the Sun, relaxation to linear force-free fields only takes place at limited occasions (e.g., in a flare) and over a limited volume which magnetic reconnection (or other processes) can propagate and homogenize the twist.